Mathematics educators are frequently confronted with the question of the use of learning Mathematics. Why should one go through all the formalisms of math when it’s very unlikely that an average person is going to use most of it in his daily life? Well sometimes it’s easy to convince people of the need for numeracy and even abstract maths. But often it’s a hard row to hoe. However, once in a blue moon, mathematics makes the headline in the news and the Nation’s attention is caught over its importance. At least for a while. Here’s a snippet from one of the leading dailies in the United States –

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One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.
Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray.

Why do Americans Stink at Math? The New York Times

Kurt VanLehn, one of the pioneers in understanding such systematic erratic thinking and working on computer models that could detect such “bugs”, wrote a book on the subject in 1985 called Mind Bugs. Today, mathematical pedagogy literature fondly calls them “misconceptions”. Whether we refer to these as bugs or Misconceptions, the key is to recognize that it’s not just about lacking the knowledge or numeracy, but having a systematic and seemingly logical reason which turns out to be incorrect. Such misconceptions have been much studied over time at a very granular level. For the sake of brevity suffice it to say that misconceptions are not just restricted to the domain of mathematics but are to be found in almost any other subject.

To begin with, there are two components to deal with misconceptions. Identifying whether someone has a misconception. And figuring out the root cause for the misconception. Misconceptions seldom arise due to incorrect teaching. They arise because of the way a concept has been taught. Thus, identifying the root cause of a misconception pedagogically helps an educator fine tune the pedagogy to ensure no gap is left for any arousal of bugs.

In fact in the latter half of 20th century there was a lot of interest in studying the unschooled population for numeracy. Unschooled population comprises of those who didn’t receive formal education. When we focus on mathematics and specifically numeracy, many studies pointed to the fact that unschooled workers were equally good if not worse compared to those who received formal education – at many numeracy tasks. After all a worker charged with the loading and unloading of carts or a kid running errands to purchase groceries would be equally good at doing the necessary calculations. There are some nuances in interpreting these studies because of subtleties like the unschooled workers not doing so well when they were given tasks to do on paper as opposed to computing mentally. However the cognitive-science research from this time almost suggested that the primary cause of numeracy could be School!

Although it seems to be counter intuitive at first, it actually makes sense when we understand the reason for the origins of such bugs. For example, a vendor calculating the price of x items would do better if he computes mentally than when asked to do it on paper. When he tries to apply what he learned at school on paper he is prone to make systematic procedural errors. This turned out to be the broader reason for misconceptions. It’s not just pure pedagogy but the underlying philosophy of mathematics education, which at that time was far too focused on imparting procedural knowledge that educators expect students to become adept at – with pure practice. Albeit the work on refining pedagogy is still in progress much has changed in the last few decades.

The National level blunder quoted above is one of the common misconceptions among school going students. Pedagogically this misconception is labelled as Among fractions with same numerators greater the denominator larger is the fraction. To address this particular misconception the change in pedagogy required is to focus on the understanding of unit fraction as part of a whole rather than simply teaching the thumb rule that larger the denominator, smaller the unit fraction. Further it should be elucidated why part-whole thinking leads to the correct comparison of fractions – the more number of parts a whole is divided into, the smaller each part is going to be. Armed with this conceptual knowledge a student need not memorize any rule or procedure to compare the fractions. Understanding a concept is proven to be a better way of retaining learning than simply memorizing.

Another common misconception prevalent among many adults is regarding negative numbers. An United Kingdom’s lottery company Camelot had to scrap off one of it’s scratchpad lotteries because of innumeracy among the participants –

The Cool Cash game – launched on Monday – was taken out of shops yesterday after some players failed to grasp whether or not they had won.
To qualify for a prize, users had to scratch away a window to reveal a temperature lower than the figure displayed on each card. As the game had a winter theme, the temperature was usually below freezing.
But the concept of comparing negative numbers proved too difficult for some. Camelot received dozens of complaints on the first day from players who could not understand how, for example, -5 is higher than -6.

“Cool Cash” card confusion, Manchester Evening News

We are already standing on the shoulders of giants. We have much literature documented on the variety of misconceptions found in many domains of mathematics and research on student’s learning is always burgeoning. It’s time the curricula carefully take care of the pedagogical implications expounded by this research body. Likewise teachers need to be trained on the cognitive aspects of misconceptions and we need to enable them to follow effective teaching methodologies.